Calculate The Product: { (2-4i)(-6+4i)$}$.

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Introduction

Complex numbers are a fundamental concept in mathematics, and they have numerous applications in various fields, including algebra, geometry, and calculus. In this article, we will focus on calculating the product of two complex numbers, specifically the product of (2−4i)(2-4i) and (−6+4i)(-6+4i). We will use the distributive property and the concept of complex conjugates to simplify the expression and arrive at the final result.

What are Complex Numbers?

Complex numbers are numbers that can be expressed in the form a+bia+bi, where aa and bb are real numbers, and ii is the imaginary unit, which satisfies the equation i2=−1i^2=-1. Complex numbers can be represented graphically on a complex plane, with the real part aa on the x-axis and the imaginary part bb on the y-axis.

The Distributive Property

The distributive property is a fundamental concept in algebra that states that for any real numbers aa, bb, and cc, the following equation holds:

a(b+c)=ab+aca(b+c)=ab+ac

We will use this property to expand the product of (2−4i)(2-4i) and (−6+4i)(-6+4i).

Expanding the Product

To expand the product, we will use the distributive property to multiply each term in the first complex number by each term in the second complex number.

(2−4i)(−6+4i)(2-4i)(-6+4i)

=2(−6+4i)−4i(−6+4i)=2(-6+4i)-4i(-6+4i)

=−12+8i+24i−16i2=-12+8i+24i-16i^2

Simplifying the Expression

Now that we have expanded the product, we can simplify the expression by combining like terms and using the fact that i2=−1i^2=-1.

−12+8i+24i−16i2-12+8i+24i-16i^2

=−12+32i−16(−1)=-12+32i-16(-1)

=−12+32i+16=-12+32i+16

=4+32i=4+32i

Conclusion

In this article, we have calculated the product of two complex numbers, (2−4i)(2-4i) and (−6+4i)(-6+4i), using the distributive property and the concept of complex conjugates. We have arrived at the final result, which is 4+32i4+32i. This result demonstrates the importance of complex numbers in mathematics and their numerous applications in various fields.

Complex Conjugates

Complex conjugates are pairs of complex numbers that have the same real part and opposite imaginary parts. In this case, the complex conjugate of (2−4i)(2-4i) is (−2+4i)(-2+4i), and the complex conjugate of (−6+4i)(-6+4i) is (−6−4i)(-6-4i). Complex conjugates are used to simplify expressions and to find the roots of polynomials.

Applications of Complex Numbers

Complex numbers have numerous applications in various fields, including:

  • Algebra: Complex numbers are used to solve equations and to find the roots of polynomials.
  • Geometry: Complex numbers are used to represent points and lines in the complex plane.
  • Calculus: Complex numbers are used to represent functions and to find the derivatives and integrals of functions.
  • Physics: Complex numbers are used to represent physical quantities, such as voltage and current, in electrical circuits.

Final Thoughts

In conclusion, complex numbers are a fundamental concept in mathematics, and they have numerous applications in various fields. In this article, we have calculated the product of two complex numbers, (2−4i)(2-4i) and (−6+4i)(-6+4i), using the distributive property and the concept of complex conjugates. We have arrived at the final result, which is 4+32i4+32i. This result demonstrates the importance of complex numbers in mathematics and their numerous applications in various fields.

References

  • "Complex Numbers" by Math Open Reference
  • "Complex Conjugates" by Khan Academy
  • "Applications of Complex Numbers" by Wolfram MathWorld

Further Reading

For further reading on complex numbers, we recommend the following resources:

  • "Complex Numbers" by MIT OpenCourseWare
  • "Complex Analysis" by Princeton University
  • "Complex Numbers and Geometry" by University of California, Berkeley

By following the steps outlined in this article, you can calculate the product of two complex numbers and arrive at the final result. We hope that this article has provided you with a better understanding of complex numbers and their numerous applications in various fields.

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Introduction

Complex numbers are a fundamental concept in mathematics, and they have numerous applications in various fields. In this article, we will answer some of the most frequently asked questions about complex numbers, including their definition, properties, and applications.

Q: What is a complex number?

A: A complex number is a number that can be expressed in the form a+bia+bi, where aa and bb are real numbers, and ii is the imaginary unit, which satisfies the equation i2=−1i^2=-1.

Q: What is the imaginary unit?

A: The imaginary unit, denoted by ii, is a mathematical concept that satisfies the equation i2=−1i^2=-1. It is used to extend the real number system to the complex number system.

Q: How do I add complex numbers?

A: To add complex numbers, you simply add the real parts and the imaginary parts separately. For example, if you have two complex numbers a+bia+bi and c+dic+di, their sum is (a+c)+(b+d)i(a+c)+(b+d)i.

Q: How do I multiply complex numbers?

A: To multiply complex numbers, you use the distributive property and the fact that i2=−1i^2=-1. For example, if you have two complex numbers a+bia+bi and c+dic+di, their product is (ac−bd)+(ad+bc)i(ac-bd)+(ad+bc)i.

Q: What is the complex conjugate of a complex number?

A: The complex conjugate of a complex number a+bia+bi is a−bia-bi. It is denoted by a+bi‾\overline{a+bi}.

Q: Why do we need complex conjugates?

A: Complex conjugates are used to simplify expressions and to find the roots of polynomials. They are also used in the calculation of complex numbers.

Q: What are some applications of complex numbers?

A: Complex numbers have numerous applications in various fields, including:

  • Algebra: Complex numbers are used to solve equations and to find the roots of polynomials.
  • Geometry: Complex numbers are used to represent points and lines in the complex plane.
  • Calculus: Complex numbers are used to represent functions and to find the derivatives and integrals of functions.
  • Physics: Complex numbers are used to represent physical quantities, such as voltage and current, in electrical circuits.

Q: Can complex numbers be used in real-world applications?

A: Yes, complex numbers have numerous real-world applications, including:

  • Electrical engineering: Complex numbers are used to represent voltage and current in electrical circuits.
  • Signal processing: Complex numbers are used to represent signals and to perform signal processing tasks.
  • Control systems: Complex numbers are used to represent the behavior of control systems.
  • Navigation: Complex numbers are used in navigation systems, such as GPS.

Q: Are complex numbers difficult to understand?

A: Complex numbers can be challenging to understand at first, but with practice and experience, they become easier to work with. It is essential to understand the basic concepts and properties of complex numbers before moving on to more advanced topics.

Q: What resources are available for learning complex numbers?

A: There are numerous resources available for learning complex numbers, including:

  • Textbooks: There are many textbooks available on complex numbers, including "Complex Numbers" by Math Open Reference and "Complex Analysis" by Princeton University.
  • Online resources: There are many online resources available, including Khan Academy, Wolfram MathWorld, and MIT OpenCourseWare.
  • Courses: There are many courses available on complex numbers, including online courses and in-person courses.

Conclusion

In this article, we have answered some of the most frequently asked questions about complex numbers, including their definition, properties, and applications. We hope that this article has provided you with a better understanding of complex numbers and their numerous applications in various fields.